## Smooth complete toric threefolds with no nontrivial nef line bundles.(English)Zbl 1141.14313

Summary: We describe all of smooth complete toric threefolds of Picard number $$5$$ with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than $$5$$.

### MSC:

 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14C20 Divisors, linear systems, invertible sheaves 14J30 $$3$$-folds
Full Text:

### References:

 [1] L. Bonavero, Sur des variétés de Moishezon dont le groupe de Picard est de rang un, Bull. Soc. Math. France 124 (1996), no. 3, 503-521. · Zbl 0866.32014 [2] L. Bonavero, Sur des variétés toriques non projectives, Bull. Soc. Math. France 128 (2000), no. 3, 407-431. · Zbl 0954.14038 [3] M. Eikelberg, The Picard group of a compact toric variety, Results Math. 22 (1992), no. 1-2, 509-527. · Zbl 0786.14031 [4] G. Ewald, Spherical complexes and nonprojective toric varieties, Discrete Comput. Geom. 1 (1986), no. 2, 115-122. · Zbl 0597.52009 [5] O. Fujino, On the Kleiman-Mori cone, Proc. Japan Acad., 81A (2005), no. 5, 80-84. · Zbl 1093.14025 [6] W. Fulton, Introduction to toric varieties , Ann. of Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993. · Zbl 0813.14039 [7] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings . I, Lecture Notes in Math., 339, Springer, Berlin, 1973. [8] P. Kleinschmidt and B. Sturmfels, Smooth toric varieties with small Picard number are projective, Topology 30 (1991), no. 2, 289-299. · Zbl 0739.14032 [9] J. Kollár, Flips, flops, minimal models, etc., in Surveys in differential geometry ( Cambridge , MA , 1990), 113-199, Lehigh Univ., Bethlehem, PA. · Zbl 0755.14003 [10] O. Nagaya, Classification of 3-dimensional complete non-singular torus embeddings, Master’s thesis, Nagoya Univ. 1976. [11] I. Nakamura, Moishezon threefolds homeomorphic to $$\mathbf{P}^{3}$$, J. Math. Soc. Japan 39 (1987), no. 3, 521-535. · Zbl 0629.14034 [12] T. Oda, Torus embeddings and applications , Tata Inst. Fund. Res., Bombay, 1978. [13] T. Oda, Convex bodies and algebraic geometry , Translated from the Japanese, Springer, Berlin, 1988. · Zbl 0628.52002 [14] K. Oguiso, Two remarks on Calabi-Yau Moishezon threefolds, J. Reine Angew. Math. 452 (1994), 153-161. · Zbl 0792.14022 [15] S. Payne, A smooth, complete threefold with no nontrivial nef line bundles, (2005). (Preprint). math.AG/0501204. · Zbl 1141.14313
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.